Confining-potential energy space (${\Sigma }_V$)

Let $V$ be a nonnegative, locally bounded measurable function on ${\mathbb{R}}^d$ with $V(x)\to \infty$ as $|x|\to \infty$. Its confining-potential energy space isThe displayed quantity is a norm even though it omits an explicit L2 norm term. Choose a ball outside which $V\ge 1$. The weighted term controls $\Vert u{\Vert }_2$ outside that ball, and a local Poincaré inequality anchored on a surrounding annulus controls the interior L2 norm by the gradient and that annular mass. Thus the displayed norm is equivalent to the full graph norm containing $\Vert u{\Vert }_2$.

A Cauchy sequence converges in $H^1$ and its weighted values converge in L2 space; an almost-everywhere convergent subsequence identifies the latter limit with $V^{1/2}u$. Hence ${\Sigma }_V$ is a Hilbert space. Growth of the potential makes bounded-energy mass tails small, which yields compact embedding of a confining-potential energy space.

The embedding ${\Sigma }_V\hookrightarrow L^2({\mathbb{R}}^d)$ is compact. The Rellich-Kondrachov compactness theorem gives compactness on each fixed ball, whilemakes the tails uniformly small.

For $\eta >0$ in two dimensions, the quadratic functionalattains its minimum subject to $\Vert u{\Vert }_4^4=M>0$. A bounded minimizing sequence is weakly compact in the harmonic-oscillator energy space and strongly compact in $L^4$: local Rellich-Kondrachov compactness theorem and the weighted tail bound prevent escape to infinity.

The minimum ofsubject to $\Vert u{\Vert }_2=1$ is attained because ${\Sigma }_V$ embeds compactly into $L^2$. Replacing a minimizer by its absolute value does not increase its energy. The Euler-Lagrange equation therefore produces a nonnegative eigenfunction of $-\Delta +V$ for its lowest eigenvalue.